In Attribute based encryption and predicate encryption, the message M which is encrypted with the ciphertext belongs to group G, GT used in the specific encryption scheme, i.e. M.e(g,g)^s where M is element of G or GT.
My question is that, is it possible for M to have a different value, i.e. M = 1,2,3,A,B etc and not an element belonging to groups G or GT.
You can always take a public key encryption and turn it into a Key Encapsulation (KEM) so it encrypts a key (rather than a message) and the key encrypted (in the group) is then fed into a Key Derivation function (KDF) to get a key from the group element represented as a bit string, mapped by the KDF into a bit string key for a symmetric encryption (say, AES), and use this key to encrypt the "user defined message" (that part is called message encapsulation). Obviously, only if you can derive the encapsulated key you can read the message, so this property (called "Hybrid Encryption") should translate to predicate encryption.
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